ACTIVITY OF A SOLUTE FROM DISTRIBUTION BETWEEN TWO IMMISCIBLE SOLVENTS

17.3 ACTIVITY OF A SOLUTE FROM DISTRIBUTION BETWEEN TWO IMMISCIBLE SOLVENTS

If the activity of a solute is known in one solvent, then its activity in another solvent immiscible with the first can be determined from the equilibrium distribution of the solute between the two solvents. As an example, let us consider an extreme situation, such as that illustrated in Figure 17.7, in which the shapes of the fugacity curves are different in two different solvents. The limiting behavior at infinite dilution, Henry’s law, is indicated for each solution. The graphs reveal that the standard states are different in the two solvents because the hypothetical l-molal solutions have different fugacities.

If the solute in solution A is in equilibrium with that in solution B, its escaping tendency is the same in both solvents. Consequently, its chemical potential m 2 at equilibrium also must be identical in both solvents. Nevertheless, the solute will have different activities in solution A and B since [Equation (16.1)]

m W 2 ¼m 2 þ RT ln a 2

and (m 8 2 ) A differs from (m 2 8 ) B . If the activity of the solute is known in one of the solvents, then the activity in the other solvent can be calculated as follows. At equilibrium

(m 2 ) A ¼ (m 2 ) B (17 :10) As

(m

2 W ) A ¼ (m 2 ) A þ RT ln (a 2 ) A

Figure 17.7. Comparison of figacity – molality curves for solute in two immiscable solvents; used to determine Henry’s law constants.

DETERMINATION OF NONELECTROLYTE ACTIVITIES AND EXCESS GIBBS FUNCTIONS

and

(m 2 ) ¼ (m W B 2 ) B þ RT ln (a 2 ) B

it follows from Equation (17.10) that (m W

A 2 B (17 :11)

(a

Therefore, to calculate (a 2 ) B from (a 2 ) A , we must find the difference between the chemical potentials in the respective standard states. From Equation (15.10)

2 ¼m 2(gas) þ RT ln 2 f W

2 ) B ¼ (k 2 ) A (17 :13)

(a 2 )

(k A 00 2 ) B

To determine the ratio of the activities in the two solvents, then, we need to determine the ratio of the Henry’s-law constants for the solute in the two solutions.

For these solutions, from Equation (15.8),

At equilibrium, ( f 2 ) A ¼(f 2 ) B , because both solutions are in equilibrium with the same vapor, and

(m )

(k 00 ) m lim 2 A ¼ 2 B (17 2 :14) !0 (m 2 ) (k 00 B 2 ) A

17.4 ACTIVITY FROM MEASUREMENT OF CELL POTENTIALS

Figure 17.8. Extrapolation of distribution data to obtain the conversion factor for solute activities.

Thus, Equation (17.13) can be written as

(m )

(a 2 ) B ¼ (a 2 ) lim

A m (17 :15) 2 !0 (m

If the activity in one solvent is known as a function of molality, and if the equili- brium molalities in both solvents can be determined for a range of molalities, Equation (17.15) can be used to calculate the activity in the other solvent. A value for

is obtained by plotting values of (m 2 ) B /(m 2 ) A at equilibrium against (m 2 ) A and by extrapolating to (m 2 ) A ¼ 0, as in Figure 17.8.

17.4 ACTIVITY FROM MEASUREMENT OF CELL POTENTIALS Although potential measurements are used primarily to determine activities of elec-

trolytes, such measurements can also be used to obtain information on activities of nonelectrolytes. In particular, the activities of components of alloys, which are solid solutions, can be calculated from the potentials of cells such as the following for lead amalgam:

Pb(amalgam, X 0 2 ); Pb(CH 3 COO) 2 , CH 3 COOH; Pb(amalgam, X 2 ) (17 :16) In this cell, two amalgams with different mole fractions of lead act as electrodes in a

common electrolyte solution containing a lead salt. The activities of lead in these amalgams can be calculated from emf measurements with this cell.

DETERMINATION OF NONELECTROLYTE ACTIVITIES AND EXCESS GIBBS FUNCTIONS

If we adopt the convention of writing the chemical reaction in the cell as occurring so that electrons will move in the outside conductor from left to right, so that the reaction in the left electrode is an oxidation and the reaction in the right electrode is a reduction, the reaction at the left electrode is

(17 :17) and that at the right electrode is

Pb(amalgam, X 0 2 2 þ ) ¼ Pb þ 2e

(17 :18) As the electrolyte containing Pb 2 þ is common to both electrodes, the cell reaction,

Pb 2 þ þ 2e ¼ Pb(amalgam, X 2 )

that is, the sum of Reactions (17.17) and (17.18), is Pb(amalgam, X 0 2 ) ¼ Pb(amalgam, X 2 )

(17 :19) As the two amalgams have the same solvent (mercury), we may choose the same

standard state for Pb in each amalgam. We choose a Henry’s-law standard state because we have data for dilute amalgams and the solubility of Pb in Hg is limited. Equation (16.18) then will represent the Gibbs function change for Equation (17.19):

a 2 DG m ¼ RT ln

If concentrations in the alloy are expressed in mole fraction units, then [Equation (16.7)]

a 2 ¼X 2 g 2

As, from Equation (7.84),

it follows that

a 0 (17 2 :20)

¼ RT ln 2 g 2

a 0 (17 2 :21) so

ln a 0 2 (17 :22)

The potentials are measured for a series of cells in which X 2 is varied and X 0 2 is held constant. A typical series of data for lead amalgam is shown in Table 17.1.

17.4 ACTIVITY FROM MEASUREMENT OF CELL POTENTIALS

395 To obtain values of a 2 or of g 2 from these data, we rearrange Equation (17.22) to

the form

ln a 0 ln g (17 :23)

2 ¼ ln a 0 2 2 (17 :24)

RT

TABLE 17.1. Electromotive Force of Pb (amalgam, X 2 0 5 6.253 3 10 24 );

Pb(CH 3 COO) 2 , CH 3 COOH; Pb (amalgam, X 2 ) at 2588888C a [3]

X 2 nF E /RT þ ln X 2 1000a 2 g 0.0006253

a M. M. Haring, M. R. Hatfield, and P. P. Zapponi, Trans. Electrochem. Soc. 75, 473 (1939).

DETERMINATION OF NONELECTROLYTE ACTIVITIES AND EXCESS GIBBS FUNCTIONS

The left side of Equation (17.24) is evaluated from the data in Table 17.1, with n ¼ 2, because two electrons are transferred per Pb transferred. Then this quantity is plotted

against X 2 and extrapolated to X 2 ¼ 0, as in Figure 17.9.

We can write a limiting form of Equation (17.24)

nF E

lim

þ ln X 2 ¼ ln a 0 2 (17 :25)

x !0

2 RT

Then the extrapolated value from Figure 17.9 is equal to ln a 2 0 . By a least-squares fit of the most dilute points, which are essentially linear, we obtain a value of 27.3983 +

0.0016. Once a 0 2 is known, a 2 can be determined at various mole fractions from Equation (17.20) in the form

The results for the lead amalgams of the cell in Equation (17.16) are assembled in Table 17.1. The activity coefficients also have been calculated. The graphical

representation of ln g 2 is shown in the body of Figure 17.9.

From Equations (16.7) and (16.17), we can write

Figure 17.9. Extrapolation of cell potential data from Table 17.1 to obtain a constant to cal- culate activities of lead in lead amalgams.

17.5 DETERMINATION OF THE ACTIVITY OF ONE COMPONENT

Figure 17.10. A plot to test whether we can use Henry’s law to define a standard state for lead in lead amalgams. Data from Table 17.1.

To test whether we really observe limiting behavior, we plot a 2 /X 2 against X 2 , as in Figure 17.10.

A rigorous test would require that we have points in very dilute solution with values of a 2 /X 2 equal to 1. We shall be satisfied with a linear extrapolation by least-squares analysis of the seven-most dilute points to an intercept of 1.0029 + 0.0033, which is equal to 1 within experimental error.